Atlas › Modular forms › Weight 6 › Level 65 › 65.6.a › 65.6.a.e

Newform orbit 65.6.a.e

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 2q^{2} + 20q^{3} + 134q^{4} + 150q^{5} - 52q^{6} + 172q^{7} - 138q^{8} + 1034q^{9} + 50q^{10} + 800q^{11} - 832q^{12} - 1014q^{13} + 2108q^{14} + 500q^{15} + 322q^{16} + 4396q^{17} + 6142q^{18} + 5304q^{19} + 3350q^{20} + 1072q^{21} + 8008q^{22} - 1140q^{23} + 4756q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 22x^{5} - 1115x^{4} + 18140x^{3} + 346736x^{2} - 3753264x - 23139596$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
13	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	2	20	150	172	800	−1014	4396	5304	−1140	−3664	16664	4488	12436	1516	212	−15612	−11896	57232	−25428	214104	60808	36024	34684	−52028	129316

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 5, 13, 65]: [1, 3, 7, 36] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 4 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(65, 6)[4]   # 65.6.a.e

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 65:6 1000